Analytical Stability Solution of a Haptic Device with theInclusion of an Operator’s Model equipped with LargeVirtual Damping and Time Delay

In this paper, the stability of discrete-time and rigid haptic devices was studied in the presence of an operatorusing Routh-Hurwitz criterion and a new stability criterion was achieved. The unit circle in the z-domain wasmapped to the left half-plane and the Routh–Hurwitz method was applied to this discrete-time system. Theanalytical solution was worked out thoroughly for the first time. Stability condition was obtained and simplifiedusing Taylor series expansion. Using this new stability criterion, we analytically found that when the dampingcoefficient of virtual wall (Bw) was increased beyond a certain level, it had a negative effect on the critical valueof the stiffness of the virtual wall (Kw). In fact, for each haptic device and user, there was a specific value of Bwwhich corresponded to the maximum achievable Kw. Equations were presented to determine the correspondingBw and Kw. The paper also provided the maximum value of Kw for some well-known haptic devices during handand finger contact with the haptic device. The results showed that for a delay of ۵ms, most haptic devices couldnot render Kw>۳kN/m in finger contact, and Kw>۵kN/m in the hand contact.

Keywords:

Haptic , Stability , Routh–Hurwitz method , Large virtual damping , Large delay